Understanding Stocks: Valuation & Growth Concepts

1. Introducing stocks Basic concepts related to the stock market The complexities of valuing stocks When to take growth opportunities Price-earnings ratios

2. An introduction to stocks • Recall that ownership in a company is through stocks • Stocks have value through… • Dividends • Being bought by another company • Any other money disbursed when the company goes out of business

3. Valuing a stock • We will see that the process of valuing a bond and most stocks is very similar • Bonds: Promised coupons or interest • Stocks: Future expected dividends • The main difference is how to determine the value of each • Bonds: You are essentially loaning money to the company • Stocks: Higher risk through ownership • You expect a higher reward (on average) due to risk

4. Firms with no dividends?!?! • Many firms have not given out dividends for years • Many of these firms plan to not give out dividends in the foreseeable future • How do these firms have value? • Reinvestment • Being bought out by another firm

5. Rapidly growing firms • Holding stock from rapidly growing firms is often riskier • Great uncertainty about future dividends • Rapidly growing firms may or may not eventually make a profit • Example: pets.com • A $300 million sock puppet Picture: From Flickr, by Jacob Bøtter from Copenhagen, Denmark

6. Growing perpetuities • Many stocks are expected to pay higher yearly dividends into the future • Growth rate over time could be constant, or could decrease once the corporation reaches a certain size • Example: 20% growth rate in dividends over the next 10 years, followed by 8% growth thereafter

7. Time horizon issues • Some people are concerned that potential dividend payments made very far into the future (>50 years) are not valued in the market • There are two ways to approach this • Discounting far into the future • Dividend vs. dividend/sales approaches to stock valuation

8. Discounting far into the future • For even moderate discount rates, payments made far into the future lead to very little of a stock’s value • Example: $5 dividend yearly forever

9. $5 dividend yearly forever • A stock only a has a small amount of its value from dividends paid out more than 50 years in the future • We are assuming a constant dividend each year • $5 dividend each year forever with r = 8% has a PV of $62.50 • The same stock received 50 years from now has PV of $1.33 Note: We assume that the first dividend payment is made one year from now

10. Dividend vs. dividend/sales approaches to stock valuation • We can also compare a stock’s value two ways to get PV • Dividend stream approach • Find the PV of expected future dividends • Dividend/sales approach • Let’s examine this…

11. Dividend/sales approach • We can value a stock into two components using the dividend/sales approach • The dividend paid one year from now ($5) • The expected amount of money we can sell the stock for one year from now ($62.50) • This incorporates all payments in the future (even those far into the future) • Note that we have to discount both future payments by one year

12. Dividend/sales approach: Some algebra • How do we know that $62.50 is the right price using the dividend/sales approach? • We can solve this algebraically • Solve x = (5 + x) / 1.08  x = 62.50 • We know that the price stays the same over time, because of the constant dividend

13. Other types of growth of dividend payments • So far, we have only looked at stocks with expected dividends that are the same each year • Other types of stocks • Constant growth • Could be positive or negative • Differential growth • In these examples, growth usually starts at a high rate, then diminishes when the company size gets big

14. Constant growth rate • We may expect a dividend to increase over time • Example • $3 dividend paid one year from now • The dividend increases by 4% each year • You have an annual discount rate of 12%

15. $3 dividend year 1,4% growth per year • $3 dividend in year 1 • $3.12 dividend in year 2 • $3.2448 dividend in year 3 • etc... • We have already done this type of perpetuity • Growing perpetuity • Dividend in year 1, divided by (R – g)

16. When the growth rate is small relative to the discount rate, a doubling of the growth rate will not affect the stock’s value much Cautionary notes • In some cases, when the growth rate doubles, the stock’s value MORE THAN doubles • Example: $10 dividend in year 1 • Assume two growth rates, 4% and 8% • Assume discount rate is 11% • If we assume that g> R, then the value of the stock is infinite • If this is the case, we have underestimated the annual discount rate of the stock, probably due to underestimating risk

17. Growing dividend example • $10 dividend in year 1 • Assume two growth rates, 4% and 8% • Assume discount rate is 11% • 4% growth • PV = $10 / (.11 – .04) = $142.86 • 8% growth • PV = $10 / (.11 – .08) = $333.33

18. Why does the value more than double? • Let’s take a look at the denominators of the PVs we calculated • PV = $10 / (.11 – .04) • Denominator is 0.07 • PV = $10 / (.11 – .08) • Denominator is 0.03 • When the denominator is cut by more than half, the PV more than doubles What does g need to be for PV to double? Solve 0.11 – x = 0.035

19. How do we value stocks? • Recall that there are three potential ways that stocks have value • Dividends • Payments made directly to stockholders • We will typically focus on stock value as a perpetuity of dividends • Being bought out by another company • Money given to stockholders after going out of business

20. Differential growth:More complicated • We do not have a nice formula for calculating the value of a stock with differential growth • Instead, we need to calculate the PV of the dividend each year of higher growth, and then the PV of the remaining payments in the future • PV of future payments is just a discounted perpetuity

21. Differential growth:Example • $1.20 dividend paid one year from now • 25% growth in the dividend through year 4 • 12% growth in the dividend thereafter • Annual discount rate is 20% • What is the PV of this stock?

22. $1.20 dividend year 1, 25% growth, followed by 12% growth • $1.20 dividend in year 1 has PV of $1 • $1.50 dividend in year 2 has PV of $1.0417 • $1.50 dividend calculated from $1.20 (1.25) • PV calculated from $1.50 / (1.2)2 • $1.875 dividend in year 3 has PV of $1.0851 • $2.3438 dividend in year 4 has PV of $1.1304 • Remaining dividends • Use the perpetuity formula and discount by 4 years Remember that the annual discount rate is 20%

23. We can consider the remaining dividends to being equivalent to a perpetuity with first payment five years from now Remember that the dividend increases by 12% between year 4 and year 5 Dividend of $2.625 in year 5 We can now apply the growing perpetuity formula to calculate the value in year 4 $2.625 / (.2 – .12) = $32.813 We must discount this by 4 years to get PV $32.813 / (1.2)4 = $15.8239 Remaining dividends

24. $1.20 dividend in year 1 PV of $1 $1.50 dividend in year 2 PV of $1.0417 $1.875 dividend in year 3 PV of $1.0851 $2.3438 dividend in year 4 PV of $1.1304 Total PV of dividends paid years 5 and on $15.8239 Total PV of all dividends $1 + $1.0417 + $1.0851 + $1.1304 + $15.8239 $20.08 (rounded to the nearest cent) Adding all of the PVs up

25. How do wedetermine g and R? • So far, we have looked at examples assuming what g and R are • We will look at how these variables are derived • Simple cases now with some (possibly unrealistic) assumptions • No issuing of stocks or bonds • Assume that some earnings are not paid out as dividends • Assume that earnings stay the same over time when all earnings are paid out as dividends

26. Determining g • Based on our assumptions we get… • Earnings next year = Earnings this year + Increase in earnings • But the increase in earnings is… • Retained earnings this year * Return on retained earnings • Notice that we want a good return for retained earnings, or else we are better off distributing this money to stockholders

27. Some algebra to help determine g • So Earnings next year is equal to • Earnings this year + Retained earnings this year * Return on retained earnings • Dividing everything by earnings this year gives us • Earnings next yr = Earnings this yr + Retained earnings this yr * Return on Earnings this yr Earnings this yr Earnings this yr retained earnings

28. Finding g • Earnings next yr = Earnings this yr + Retained earnings this yr * Return on Earnings this yr Earnings this yr Earnings this yr retained earnings • We call the ratio in bold the retention ratio • In other words, how much of this year’s earning is retained for investing • Simplifying the above equation gets us • 1 + g = 1 + Retention ratio * Return on retained earnings • g = Retention ratio * Return on retained earnings

29. Example • A company earns $1 million over the next year • Note that future earnings will depend on the amount of retained earnings here • 70% of earnings are paid out as dividends • 30% of earnings are retained • Return on retained earnings is 14% • g = Retention ratio * return on retained earnings = .3 * .14 = .042 (Note that both earnings and dividends grow at 4.2%)

30. Determining R • Let P be the price in year 0 • P = Div / (R – g) • We can solve for R • R – g = Div / P • R = (Div / P) + g dividend yield capital gains yield • So the discount rate has two components • Dividend yield • Capital gains yield

31. When should a firm retain earnings? • When is it in a firm’s best interest to retain earnings? • Again, we assume that a company cannot issue stocks or bonds for now • As always, we want a positive NPV for a project • We assume a discount rate of R when determining the NPV of a project

32. Application: Cash cowvs. reinvestment • If a company reinvests none of its earnings, it is referred to as a cash cow • The value of the stock is simply Div / R • Note that earnings per share is the same as the dividend, due to no reinvesting • A stock’s value with reinvestment has two components • Value with no reinvestment • Earnings per share divided by R • NPV of the investment (sometimes referred to as a growth opportunity) • Shorthand notation: NPVGO

33. Example:New project • If a firm acts as a cash cow, the yearly dividend is $4 • The firm’s discount rate is 8% • Firm’s value is $4 / (0.08) = $50 • The firm could retain $1 of earnings every year into a project that will earn $12.32% the following year • Should the firm invest? Two ways to think about it • YES! NPV of investment is positive • YES! Rate of return is higher than discount rate

34. Invest? YES! • The firm should invest, since the rate of return on the investment exceeds the discount rate • Investing one dollar this year leads to $1.1232 one year from now • PV is $1.04 once we discount one year at 8% • NPV = $1.04 – $1 = $0.04 • NPV of this investment made year after year is $0.04 + ($0.04 / 0.08) = $0.54 • Value of stock with reinvestment is • $50 + $0.54 = $50.54

35. Price-earnings ratio • Let EPS = earnings per share • Then Price per share = (EPS / R) + NPVGO • If we divide both sides by EPS, we get • Price per share / EPS = (1 / R) + (NPVGO / EPS) Price-earnings ratio • Notice that if two firms each report $1 in earnings per share of stock, the stock of the firm with better growth opportunities will have a higher price-earnings ratio (assume R is the same for both firms) • NPVGO / EPS is higher for the firm with better growth opportunities

36. A final thought on reinvestment • Note that marginal returns are usually decreasing • Thus, a firm that reinvests part of its earnings will not necessarily increase its value if it continues to reinvest more • Firms will often reinvest in order to maximize stock value

37. This concludes Unit 2 • In the coming lectures (Unit 3) • What is the relationship between risk and return? • Can we come up with a formal model? • What is the history of risky asset returns? • How did the year 2008 change perceptions of risk?

38. Next lecture: Moreon stocks and risk